Difference between revisions of "Singleton (set theory)"
From Maths
(Created page with "__TOC__ =={{subpage|Definition}}== {{/Definition}} More concisely this may be written: * {{M|\exists t\in X\forall s[s\in X\implies t\eq s]}} ==Significance== Notice that we h...") |
(Updating after mistake spotted in definition, thank goodness I prove stuff!) |
||
| Line 3: | Line 3: | ||
{{/Definition}} | {{/Definition}} | ||
More concisely this may be written: | More concisely this may be written: | ||
| − | * {{M|\exists t\in X\forall | + | * {{M|\exists t\in X\forall s\in X[t\eq s]}} (see [[rewriting for-all and exists within set theory]]) |
| + | ** For proof see '''Claim 1'''. | ||
==Significance== | ==Significance== | ||
Notice that we have manage to define a set containing one thing without any notion of the number 1. | Notice that we have manage to define a set containing one thing without any notion of the number 1. | ||
==See next== | ==See next== | ||
* [[A pair of identical elements is a singleton]] | * [[A pair of identical elements is a singleton]] | ||
| + | ==Proof of claims== | ||
| + | # {{M|\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)}} | ||
| + | #* By "''[[rewriting for-all and exists within set theory]]''" we see: | ||
| + | #** {{M|\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)}} | ||
| + | #**: {{M|\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)}} by simplification | ||
| + | #**: {{M|\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)}} by changing the bracket style | ||
| + | #**: {{M|\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)}} by re-writing again. | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Elementary Set Theory|Set Theory}} | {{Definition|Elementary Set Theory|Set Theory}} | ||
Revision as of 17:35, 8 March 2017
Definition
Let [ilmath]X[/ilmath] be a set. We call [ilmath]X[/ilmath] a singleton if[1]:
- [ilmath]\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)][/ilmath]Caveat:See:[Note 1]
- In words: [ilmath]X[/ilmath] is a singleton if: there exists a thing such that ( the thing is in [ilmath]X[/ilmath] and for any stuff ( if that stuff is in [ilmath]X[/ilmath] then the stuff is the thing ) )
More concisely this may be written:
- [ilmath]\exists t\in X\forall s\in X[t\eq s][/ilmath][Note 2]
More concisely this may be written:
- [ilmath]\exists t\in X\forall s\in X[t\eq s][/ilmath] (see rewriting for-all and exists within set theory)
- For proof see Claim 1.
Significance
Notice that we have manage to define a set containing one thing without any notion of the number 1.
See next
Proof of claims
- [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)[/ilmath]
- By "rewriting for-all and exists within set theory" we see:
- [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)[/ilmath]
- [ilmath]\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)[/ilmath] by simplification
- [ilmath]\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)[/ilmath] by changing the bracket style
- [ilmath]\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)[/ilmath] by re-writing again.
- [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)[/ilmath]
- By "rewriting for-all and exists within set theory" we see:
Notes
- ↑ Note that:
- [ilmath]\exists t[t\in X\rightarrow\forall s(s\in X\rightarrow s\eq t)][/ilmath]
- ↑ see rewriting for-all and exists within set theory
